### Article

## A pointwise selection principle for maps of several variables via the total joint variation

Given a rectangle in the real Euclidean n-dimensional space and two maps f and g defined on it and taking values in a metric semigroup, we introduce the notion of the total joint variation TV(f , g) of these maps. This extends similar notions considered by Hildebrandt (1963) [17], Leonov (1998) [18], Chistyakov (2003, 2005) [5,8] and the authors (2010). We prove the following irregular pointwise selection principle in terms of the total joint variation: if a sequence of maps {fj}∞ j=1 from the rectangle into a metric semigroup is pointwise precompact and lim supj,k→∞ TV(fj, fk) is finite, then it admits a pointwise convergent subsequence (whose limit may be a highly irregular, e.g., everywhere discontinuous, map). This result generalizes some recent pointwise selection principles for real functions and maps of several real variables.

Given a closed interval $I=[a,b]$ and a metric space $(M,d)$, we introduce a

nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^I$ (the set of all

functions from $I$ into $M$), called the {\it joint modulus of variation}. We show that

if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^I$ are such that

$\{f_j\}$ is pointwise relatively compact on $I$, $\{g_j\}$ is pointwise convergent on $I$,

and $\limsup_{j\to\infty}\nu_n(f_j,g_j)=o(n)$ as $n\to\infty$, then $\{f_j\}$ admits

a pointwise convergent subsequence whose limit on $I$ is a conditionally regulated function.

Given two points *a*=(*a*_{1},…,*a**n*) and *b*=(*b*_{1},…,*b**n*) from *R**n* with *a*<*b* componentwise and a map *f* from the rectangle into a metric semigroup *M*=(*M*,*d*,+), we study properties of the *total variation* of *f* on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217–222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (*n*=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (*n*=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61–71] (*n*∈*N*) for real-valued functions of *n* variables.

In this paper we consider sequences of functions that are defined on a subset of the real line with values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent research. In addition, we prove that the sufficient condition is also necessary for uniformly convergent sequences of functions. We also obtain a representation for regular functions whose values belong to the uniform space.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.